susan friedlander and victor yudovich- instabilities in fluid motion

Instabilities in FluidMotionSusan Friedlander and Victor Yudovich

—“The diversity of problems exhibited in hydrodynamics alone can only make one admire theinexhaustibility of fascinating problems that occur in nature.”

—Birkhoff, Bellman, and Lin, from Hydrodynamic Instability, 1962.

1358 NOTICES OF THE AMS VOLUME 46, NUMBER 11

Our everyday life is full of examples offluid motion, from the drama of tor-nadoes and hurricanes, the turbulentcascading flows in rivers, and thebreaking of massive ocean waves to

the undramatic familiarity of stirring a cup of tea.Inspired by observation, scientists have soughtfor centuries to understand and predict fluid be-havior. Over two hundred years ago Euler formu-lated his celebrated equations that give the fun-damental mathematical description of fluid motion.It is interesting to note that other areas of physicssuch as the theory of heat or the motion of wavesadvanced rapidly in the early nineteenth centurybecause linear models were in good agreementwith experiments and the mathematical tool ofFourier series was successfully developed to attacksuch linear equations. While almost all the partialdifferential equations in mathematical physics arenonlinear, a number such as the heat equationand the wave equation become linear in commonlyoccurring physical situations (e.g., the heat equa-tion when the coefficient of thermal conductivityis independent of the temperature itself). In con-trast, the process of understanding fluid behaviorwas much slower because of the crucial nonlin-earity in the Euler equations.

Only the simplest flows in which all fluid par-ticles are moving along straight lines or around cir-cles could be described by explicit solutions ofthe hydrodynamic equations. The theoretical so-lutions named Poiseuille flow in a circular pipe androtating Couette flow between concentric circularcylinders were found in the mid-nineteenth cen-tury. However, by the end of the nineteenth cen-tury a striking experimental fact became clear:these flows were realizable in experiments onlywhen they were sufficiently slow. Reynolds [1]gave a beautiful description that we reproduce

Susan Friedlander is professor of mathematics at the Uni-versity of Illinois at Chicago. Her research is partiallysupported by NSF grant DMS-99 70977. Her e-mail addressis [email protected].

Victor Yudovich is professor of mathematics at the Uni-versity of Rostov. His research is partially supported byRussian grant RFBR 96-15-96081. His e-mail address [email protected].

Figure 1. Diagram of the experimentalapparatus used by Reynolds. After Reynolds(1883), see [1].

fea-friedlander.qxp 10/15/99 11:15 AM 358

DECEMBER 1999 NOTICES OF THE AMS 1359

below of an experiment showing that as the speedof water in a pipe is increased, the flow completelychanges its nature from simple Poiseuille flow toa completely different regime that is very irregu-lar in time and space. Reynolds called this turbu-lence and pointed out that the transition from thesimple flow to the chaotic flow was caused by thephenomenon of instability.

The issue of the stability or instability of a fluidflow became one of the most basic problems influid dynamics and was examined experimentallyand mathematically by such giants of science asHelmholtz, Kelvin, Rayleigh, and Reynolds. It pre-sents an important example of a physical problemthat can be partially addressed through sophisti-cated mathematics and where the answers have di-rect physical interpretations: stable flows are ro-bust under inevitable disturbances in theenvironment, while unstable flows may break up,sometimes rapidly. Reynolds describes a simple ex-periment that illustrates a progression from sta-ble to unstable to turbulent fluid behavior:

The …experiments were made on threetubes…. The diameters of these werenearly 1 inch, 12 inch, and 14 inch. Theywere all…fitted with trumpet mouth-pieces, so that the water might enterwithout disturbance. The water wasdrawn through the tubes out of a largeglass tank, in which the tubes were im-mersed, arrangements being made sothat a streak or streaks of highlycoloured water entered the tubes withthe clear water.

The general results were as follows:

(1) When the velocities were suffi-ciently low, the streak of colour ex-tended in a beautiful straight linethrough the tube.

(2) If the water in the tank had notquite settled to rest, at sufficiently lowvelocities, the streak would shift aboutthe tube, but there was no appearanceof sinuousity.

(3) As the velocity was increased bysmall stages, at some point in the tube,always at a considerable distance fromthe trumpet or intake, the color bandwould all at once mix up with the sur-rounding water, and fill the rest of thetube with a mass of coloured water.Any increase in the velocity caused thepoint of breakdown to approach thetrumpet, but with no velocities thatwere tried did it reach this. On viewing

the tube by the light of an electric spark,the mass of colour resolved itself intoa mass of more or less distinct curls,showing eddies.

As we mentioned, the stability/instability of theflow in the pipe is governed in Reynolds’s experi-ment by increasing the velocity. In fact, the full con-trol parameter is the dimensionless number nowknown as the Reynolds number,

R = Ud/ν,

where U is the mean speed of the fluid, d is thediameter of the pipe, and ν is the viscosity of thefluid. In Reynolds’s series of experiments, U wasincreased: alternatively one could envision de-creasing the viscosity ν. The limit of vanishingviscosity (i.e., R →∞) is a very subtle one for thepartial differential equations (PDE) of fluid motion,and we will return to this point later.

In the hundred years that followed Reynolds’sseminal investigation of fluid instabilities, there de-veloped an enormous body of literature on thesubject. This includes not only papers in mathe-matical journals but also a large body of work inthe engineering, physics, astrophysics, oceano-graphic, and meteorological literature. It is beyondthe scope of this article to even touch on this ex-tensive bibliography, but the interested readermay find much background material in the sub-stantive texts of Lin (1955), Chandrasekhar (1961),Joseph (1976), Drazin and Reid (1981), Swinney andGollub (1981), who have contributed much to var-ious aspects of the theory of fluid stability throughtheir papers and books. An extensive bibliographycan be found in the recent volume on hydrody-namics and nonlinear instabilities edited by Go-drèche and Manneville [2]. As well as different as-pects of the physics of the problem, differentmathematical tools are used to tackle the equationsgoverning fluid stability. For example, an elegantmathematical treatment of nonlinear stability is

Figure 2. Reynolds’s sketches of the flow observed inhis 1883 experiment. (a) shows stable flow, (b) and (c)show unstable and turbulent flows.

Sou

rce:

Rey

nold

s’s

pap

er, s

ee [

1].

fea-friedlander.qxp 10/15/99 11:15 AM Page 1359


given by Arnold (1966), which is applicable tocertain two-dimensional inviscid fluid motions.An extensive discussion of the power of “energytheory” as applied to nonlinear stability of fluidflows is given by Galdi and Padula (1990).

Partly because there exist few known solutionsto the nonlinear partial differential equations thatgovern fluid motion, it is very difficult to obtaingeneral results and give detailed analysis of fluidinstabilities for arbitrary flows. Much work hasconcentrated on a relatively small number of ratherspecial fluid configurations, and even in thesecases open questions remain. For example, theReynolds experiment of 1883 is not yet fully ex-plained by current theory. Although there is no rig-orous proof of the stability of Poiseuille flow in acircular pipe, analytical and numerical evidence todate for such a flow suggests (theoretical) stabil-ity for all Reynolds numbers. As we have described,experiments show instability for sufficiently largeReynolds numbers. The resolution of this paradoxappears to be the instability of such flows with re-spect to small but finite disturbances combinedwith their stability to infinitesimal disturbances.One can conjecture that the area of attraction inphase space is contracting to one point (thePoiseuille flow itself) as R →∞. However, the proofof this conjecture is one of the challenges of hy-drodynamic stability theory.

Two Examples of InstabilitiesTo illustrate the type of physics involved, we willgive a brief description of two types of instabili-

ties involving specific, rather simple basic flowpatterns, namely, particles moving in straight linesor circles.Example 1. Instability in R2 : Kelvin–HelmholtzInstability.Consider two streams of inviscid (i.e., frictionless),incompressible fluids flowing in parallel one abovethe other with different velocities, say, one posi-tive and one negative. At the interface where thereis a discontinuity in the velocity, the vorticity (orcurl of the velocity) is nonzero. The vorticity is adelta-function-like distribution that can be mod-eled by a vortex sheet. Consider a small sinusoidaldisturbance of this sheet. Vorticity is an importantconcept in fluid dynamics, and for two-dimen-sional flows such as the one we are considering,it is conserved under the motion of the fluid par-ticles. This conservation property for an inviscidfluid implies that the vorticity in parts of the sheetdisplaced upwards (or downwards) induces a ve-locity in the positive (or negative) direction. At theundisturbed points of the sine wave, the vorticityinduces a rotational velocity about such pointsthat amplifies the sine wave; i.e., the disturbancegrows in magnitude.

It is observed that the sheet develops vortex rollsand may eventually break up in a turbulent fash-ion. The basic mechanism of this instability will beaffected by physical forces that may be presentsuch as gravity, surface tension, or frictional effectsof viscosity. Such forces may enhance (i.e., desta-bilize) or damp (i.e., stabilize) the Kelvin–Helmholtzinstability. For example, suppose there is a bottom-

Figure 3. Kelvin-Helmholtz instability of superposed streams. The upper stream of water is moving to the rightfaster than the lower one, which contains dye that fluoresces under illumination by a vertical sheet of laser

light. The faster stream is perturbed sinusoidally at the most unstable frequency in the upper photograph, andat half that frequency in the lower one so that the motion locks into the subharmonic.

Photo

grap

hs

by

F. A

. R

ob

erts

, P.

E.

Dim

ota

kis

, an

d A

. R

osh

ko.

Rep

rin

ted

wit

h p

er-

mis

sion

fro

m “

An

Alb

um

of

Flu

id M

oti

on

”, T

he

Para

boli

c Pr

ess,

1982.



heavy density distribution in the two layers. Thenfor instability to occur the velocity difference in thetwo layers must be sufficiently strong to over-come the stabilizing effects of gravity. We remarkthat Kelvin–Helmholtz type instabilities may oftenoccur in nature, and one such manifestation fa-miliar to all travelers by plane is so-called “clearair turbulence”.2. An example of instability in R3 : CentrifugalTaylor–Couette Instabilities.Consider the motion of a fluid between two con-centric cylinders rotating in the same directionwith different angular velocities. For steady motionin the absence of other forces, the centrifugal forceat any radius must be balanced by the pressure gra-dient. Consider a small radial displacement of a ringof fluid. Conservation of angular momentum im-plies a change in the angular velocity, which nowmay or may not be sufficient to offset the cen-trifugal force. If so, the displaced ring moves backto its original position and a wave is set up. If not,the ring moves away from its original position andthe disturbance is enhanced (i.e., there is instabil-ity). From such an argument a necessary and suf-ficient condition for instability to axisymmetricdisturbances can be deduced in terms of the ra-dial derivative of the angular velocity of the fluid.Experiments with a viscous fluid in which the an-gular velocity of the inner cylinder is gradually in-creased show that above a certain critical inner ro-tation rate the flow becomes unstable and onesees a pattern of small counterrotating Taylor vor-tices superimposed on the basic flow. There fol-lows a hierarchy of successive instabilities: az-imuthal travelling waves, twisting regimes,quasiperiodic regimes, and so on, which lead stepby step to developed turbulence. The mathemati-cal challenge after understanding the primary in-stability is that of studying bifurcations and de-tecting the secondary, tertiary, etc., regimes. Detailsconcerning bifurcation theory and fluid behaviorcan be found in the book of Chossat and Iooss [3].

The Euler Equations for the Motion of anIdeal FluidWe now turn to the mathematical description offluid motion. The equations that describe the mostfundamental behavior of a fluid were derived byEuler in 1755. They are the equations of conser-vation of momentum and conservation of mass ofa fluid that is incompressible, has constant den-sity, and is inviscid. Such a fluid is sometimescalled a “perfect” or an “ideal” fluid. Let q(x, t) de-note the velocity vector and P (x, t) denote thescalar pressure field. The Euler equations are thefollowing system of nonlinear PDEs:

(1)∂q∂t

+ (q · ∇)q = −∇P

(2) ∇ · q = 0.

Equation (1) is the mathematical formulation ofNewton’s second law of motion written for an el-ement of an ideal fluid. Equation (2) represents con-servation of mass for an incompressible fluid.Heuristic derivations of these equations can befound in many texts, including those of Batchelor(1967) and a very readable introduction to fluid dy-namics by Acheson (1990). From a physical pointof view we are most interested in the case in whichq is a time-dependent vector field that takes its val-ues in R3; however, it also makes some sense toconsider motions restricted to a plane. In order topredict the motion of the fluid at time t in termsof the initial configuration, it is necessary to solvethe system (1)–(2) with given initial condition

(3) q(x,0) = q0(x)

and appropriate boundary conditions. The latterinclude such physically reasonable conditions as(a) rigid boundaries with no flow perpendicular tothe boundary or; (b) flow on a manifold without aboundary such as a torus, where q(x, t) and P (x, t)are spatially periodic; (c) flow in an infinite domainwith a condition of decay at infinity to give finiteenergy. In order to determine the motion of an ac-tual fluid particle given the velocity vector q(x, t) ,it is necessary to solve the Cauchy problem

x = q(x, t)

with

x

∣∣∣∣∣t=0

= a

Figure 4. Taylor-Couette experimental set-up.The flow regime shown here corresponds totoroidal Taylor vortices. After Schlichting(1979).

Sou

rce,

Sch

lich

tin

g, “

Bou

nd

ary

Laye

r T

heo

ry”,

1979, M

cGra

w-H

ill,

New

York

.



where a denotes the initial position of the fluid par-ticle. This system may give rise to another type ofinstability known as Lagrangian turbulence, whichis the irregular, chaotic motion of the particlesthemselves.

The initial-boundary-value problem for the Eulerequations is surprisingly difficult. Perhaps it isone of the most challenging of all problems in PDEthat arises directly from physics. Even the basicquestions of uniqueness and existence for all timeof the solutions in three dimensions remain open.More precisely, it is possible to prove uniquenessfor smooth solutions to the three-dimensionalEuler equations. However, to date, there is no proofof global existence, even of generalized solutions,to these equations. It is a great challenge to answerthe following question: Does an initially smooth ve-locity field continue to be smooth for all time asit evolves under the Euler equations? In two di-mensions the answer is yes. But in three dimen-sions the question remains highly controversial.Some mathematicians and physicists believe thatsingularities will arise from an initially smooth ve-locity field in a finite time. But no unambiguousexamples to support this contention have beenfound, and no theorems have been proved to ruleout the possibility of such a singularity. This ques-tion is important in understanding how flows be-come turbulent.

All real fluids are at least very weakly viscous.Viscosity is necessary to generate flows, and its in-fluence is very complicated. In particular, not onlyis it able to smooth and stabilize fluid motions butit also sometimes actually destroys and destabi-lizes. Incorporation of the effects of viscosity (orfriction) leads to versions of (1)–(2) known as theNavier-Stokes equations. Since friction is a fact ofnature, it could be argued that only the Navier-Stokes equations are physically relevant. But thereis much to learn from the Euler equations.

Loosely speaking, all possible fluid flows can beclassified into three kinds. The first are slow or veryviscous flows. Usually these flows go to a singleglobally stable stationary flow (i.e., the attractorconsists of a single stationary point). The secondis the class of flows with moderately large Reynoldsnumbers (small, but not very small viscosity). It isin this class that such phenomena can be foundas transitions, successive bifurcations connectedwith loss of stability, stochastic oscillations, andgrowth of the dimension of hydrodynamical at-tractors (regimes of the Taylor-Couette instabilitygive examples of this behavior). The third class isdeveloped turbulence. Figure 5 illustrates turbu-lence developing at Reynolds number 10,000. Whilemost fluid dynamicists believe that turbulenceobeys some universal laws and admits a moreexact description, as yet such laws are unknown.It is likely that the core of any such theory lies inthe asymptotics of the limit of vanishing viscos-ity, and the first step towards constructing suchan asymptotic theory is the study of inviscid flu-ids governed by the Euler equations. It is an in-teresting, and somewhat surprising, observationthat zero viscosity limit results are often also ap-plicable and consistent with experiments for flowsof the second type, i.e., moderate viscosity.

The limit of vanishing viscosity, which is the ap-propriate limit in many physical problems, is wellknown to be a subtle, singular limit. There aremany open questions connected with this limit; forexample, it is not known how regular or irregularthe limiting inviscid flows are. This is why it is nec-essary to examine all possible ideal flows in thehope that this knowledge will illuminate some gen-eral laws. Constantin [4] argues in connection withturbulence, which requires large gradients in thelimit of vanishing viscosity, that the possible de-velopment of singularities in finite time in solutionsto the Euler equations rather than the Navier–

Figure 5. Instability of an axisymmetric jet. A laminar stream of air flows from a circular tube at Reynoldsnumber 10,000 and is made visible by a smoke wire. The edge of the jet develops axisymmetric oscillations,rolls up into vortex rings, and then abruptly becomes turbulent.

Photo

grap

h b

y R

ob

ert

Dru

bka

and

Has

san

Nag

ib. R

epri

nte

dw

ith

per

mis

sion

fro

m “

An

Alb

um

of

Flu

id M

oti

on

”, T

he

Para

boli

cPr

ess,

1982.



Stokes equations is the physically relevant prob-lem.

The mathematical challenges raised by the Eulerequations are different and sometimes moredifficult than those connected with theNavier–Stokes equations. In some general sense theNavier-Stokes equations are parabolic. For certaintransitions in the Navier-Stokes equations, center-manifold techniques permit the phase space to betreated as finite dimensional, although the di-mension of the attractor for realistic values of theReynolds number is very large compared with theAvogadro number in molecular physics. However,the phase space dynamics of the Euler equation istruly infinite dimensional. The Euler equations areoutside the conventional classification of PDE the-ory, and they manifest a unique individuality. AsArnold (1966) and Ebin and Marsden (1971) demon-strated, the Euler equations are a particular caseof equations of geodesics on an infinite dimen-sional Lie group.

There are many interesting ways in which vis-cosity plays a role in the stability or instability offluid motion, and there are a number of beautifulresults concerning viscous instability theory. How-ever, in order to keep this short article focused, wewill discuss only instabilities in the Euler equationsfor the motion of an inviscid, incompressible fluid,namely, equations (1)–(2). In the context of theEuler equations Arnold [5] observed that “there ap-pear to be an infinitely great number of unstableconfigurations.” Recent progress in the study offluid instabilities bears out this observation. We willdescribe certain of these recent results and putthem in the context of some of the tools used tostudy instabilities.

Mathematical Definitions ofStability/InstabilityInstability occurs when there is some disturbanceof the external or internal forces acting on thefluid. In practice it is very hard to suppress all pos-sible small disturbances, and, loosely speaking,the question of stability or instability is the ques-tion of whether there exist disturbances that “grow”as time progresses. Mathematically the most im-portant concept of a stable steady state has its ori-gin in Lyapunov stability theory, which was firstdeveloped for ordinary differential equations (i.e.,for systems with a finite number of degrees offreedom) and then generalized to infinite-dimen-sional dynamical systems. The steady state de-scribed by a velocity field U0(x) is called Lyapunovstable if every state U(x, t) “close” to U0(x) at t = 0stays close for all t > 0. In mathematical terms“close” is defined by considering metrics in anormed space X. While in finite-dimensional sys-tems the choice of the norm is not significant be-cause all Banach norms are equivalent, in infinite-dimensional systems like a fluid configuration this

choice is crucial, as was emphasized by Yudovichin 1974.

The following general definition of stability isgiven in the book of Yudovich [6]. Consider adynamical system whose phase space is a Banachspace X. Examples for X are the space C of boundedcontinuous functions, the usual Lp space, and theSobolev space Wk,p of functions having (weak) de-rivatives through order k in Lp . We assume that ifthe initial value u0 ∈ X is given, the future evolu-tion u(t), t > 0, of the system is uniquely defined.This means there is a mapping Mt sending the ini-tial value u0 into a space M of vector functions withvalues in X:

u(t) =Mtu0.

Let us assume that zero is a steady state, i.e.,Mt (0) = 0. (We note that it is always sufficient toconsider the stability of the zero state because wecan pass to this from disturbances about an arbi-trary steady state.) The fixed point u0 = 0 is called(X,M) stable if the mapping Mt is continuous atthe point zero. For example, if the space M isC([0,∞), X

), then (X,M) stability, as defined

above, gives classical Lyapunov stability in the Ba-nach space X. If M is the subspace of vector func-tions in C

([0,∞), X

)that go to zero as t →∞, we

obtain the definition of asymptotic stability in theBanach space X.

To illustrate the dependence of stability on thechoice of norm, let us consider the following sim-ple example given in [6]. We consider the Cauchyproblem

∂u∂t

= x∂u∂x

with initial conditions

u(0) = φ(x).

The unique solution for an arbitrary smooth func-tion φ(x) is

u(x, t) = φ(xet ).

A simple calculation leads to the formula∥∥∥∥∥∂ku(·, t)∂xk

∥∥∥∥∥Lp(R)

= e(k−p−1)t‖φ(k)‖Lp(R).

Hence we see that this linear equation is:• asymptotically stable in Lp(R) for 1 ≤ p <∞

and that disturbances decay exponentially ast →∞;

• Lyapunov stable, but not asymptotically sta-ble in the Banach spaces L∞(R) , C(R) andW1,1(R);

• exponentially unstable in any space Wk,p(R)with k > 1, p ≥ 1 or k = 1, p > 1.



The Linearized Euler EquationsWe now return to the question of stability or in-stability of a specific system, namely, the Eulerequations (1)–(2) for an ideal fluid. Since we are in-terested in the behavior of fluid motion close to agiven steady flow U0(x), it is informative to writethe flow as the sum of U0 and a perturbationv(x, t) :

q(x, t) = U0(x) + v(x, t)(4)

P (x, t) = P0(x) + p(x, t).(5)

We assume that the time-dependent perturbationsv and p are “small” with respect to some metric.From the Euler equations (1) and (2), the steadyfields must satisfy:

(6) (U0 · ∇)U0 = −∇P0, ∇ ·U0 = 0.

We note that although there are many classes ofvector fields U0(x) that satisfy (6), not every vec-tor field is an equilibrium fluid flow. Neglectingterms in (1) that are quadratic in v(x, t) gives anapproximate linear system of PDEs for the evolu-tion of the perturbation velocity

(7)∂v∂t

= −(U0 · ∇)v− (v · ∇) ·U0 −∇P ≡ Lv

(8) ∇ · v = 0

with initial condition v(x,0) = v0(x) and appropri-ate boundary conditions. Equations (7)–(8) are thelinearized Euler equations for infinitesimal dis-turbances, and issues of linear stability concern thenature of the spectrum of the operator L for a givensteady velocity U0(x). From the point of view ofspectral theory for PDEs, the operator L is very non-standard, since it is nonselfadjoint and nonellip-tic. For most flows U0, none of the standard the-orems from spectral theory of PDE and ODE canbe invoked to study the spectrum1 of the opera-tor L associated with U0. Rather, the problem fallsin the context of general operator and semigrouptheory, which provides only a framework for prov-ing specific theorems.

Because of the nonstandard nature of the op-erator L, the spectrum SpecL is generically theunion of a continuous part, where f(x) may be ageneralized function, and a set of discrete eigen-values, where f(x) is an eigenfunction. If the in-tersection of Spec L with the right half of the com-plex plane is nonempty, then the linearized Eulerequation has a solution that grows exponentiallywith time. The steady flow U0(x) is then called lin-early unstable. We note that the structure of SpecLwill depend on the Banach space of functions inwhich we seek solutions to (7)–(8). An interestedreader may find refinements and further mathe-

matical definitions of stability and instability insome of the referenced texts.

Plane Parallel Shear FlowOne of the very simplest steady fluid flows U0(x)is plane parallel shear flow in which the particlesmove in straight lines with a velocity that mayvary in magnitude from level to level. In Cartesiancoordinates x1, x2, x3 in R3, we write

(9) U0(x) = (F (x2),0,0).

We assume that we have periodic boundary con-ditions and that F is C∞-smooth. In this particu-lar example the unstable spectrum is purely dis-crete except for possible limit points, and thespectral problem reduces to determining the eigen-values of an ordinary differential equation knownas the Rayleigh equation for eigenfunctions Φ(x2):

(10)(F (x2)− iσ )

[d2

dx22− k2

]Φ(x2)

−F ′′(x2)Φ(x2) = 0,

where k is in Z and Φ(x2) satisfies periodic bound-ary conditions at x2 = 0, 2π . Construction of an“energy integral” obtained by multiplying (10) bythe complex conjugate Φ∗(x2), integration by parts,and use of the boundary conditions yield the fa-mous Rayleigh Criterion, namely, that a sufficientcondition for linear stability with respect to the en-ergy metric is no inflection points in the functionF (x2) in [0,2π ] (i.e., plane parallel shear flow withno inflection points is an example of a flow thatis stable in L2 to infinitesimal disturbances).

Recently Belenkaya, Friedlander, and Yudovich(1999) used the method of averaging to prove thatfor certain profiles F (x2) where there are many in-flection points, namely rapidly oscillating profiles,the converse is true and there always exist unsta-ble eigenvalues. For the case of sinusoidal pro-files, techniques using continued fractions wereemployed to construct the complete unstable spec-trum. It is interesting, and in some sense remark-able, that the particular case of sinusoidal pro-files leads to a nonconstant coefficient eigenvalueODE (equation (10)), where it is possible to con-struct in explicit form the transcendental “char-acteristic” equation that relates the eigenvalues σand the wave numbers. Usually this can be doneonly for constant coefficient situations. In thisspecial problem a Fourier series representationfor the eigenfunctions leads to a tridiagonal infi-nite matrix for the algebraic system satisfied by theFourier coefficients. This system leads to two re-currence relations for the ratio of the coefficientsthat can be represented by infinite continued frac-tions. Equating the two continued fractions givesthe explicit form of the characteristic equation.Classical theorems from the theory of continuedfractions, whose use in fluid dynamics might be a

1By the spectrum of L we mean the set {σ ∈ C} suchthat v(x, t) = f(x)eσ t satisfies (7)–(8) for some f(x) 6= 0 .



little unexpected, can be used to prove the existenceof roots to the characteristic equation withReσ > 0 and the convergence of the Fourier series.This was first done for the viscous problem by Me-shalkin and Sinai (1961) and for equation (10) byFriedlander, Strauss, and Vishik [7].

The following simple example of Yudovich [8]can be used to illustrate the importance of thenorm in which possible growth is measured. As wenoted above, plane parallel flow with no inflectionpoint in the profile is linearly stable when thegrowth of a perturbation is measured by the en-ergy. Consider the following three-dimensionalvector:

(11) q(x, t) = (F (x2), 0, W (x1 − t F (x2))).

It is easy to check that this is an exact solution, with∇P ≡ 0, to the full nonlinear Euler equations (1)–(2)for any functions F and W . In particular, we cantake the initial value of W to be small in any met-ric on the velocity and view (11) as a small per-turbation of the shear flow (9). As we have men-tioned, a very important concept in fluid motionis the vorticity ω(x, t) = curl q(x, t) , which mea-sures the vortex-like motions in the flow. The curlof (11) is the vector

(12)∇× q = −(tF ′(x2)W ′(x1 − t F (x2)),

W ′(x1 − t F (x2)), F ′(x2)).

Hence the vorticity of such a perturbation growslinearly with time provided only that F and W arenonconstant functions. Thus, if we measure growthin any norm that incorporates the magnitude of thevorticity, we find that no matter how small the per-turbation W is initially, we have growth in time withrespect to this “vorticity” metric. Hence, in thissense even a shear flow with no inflection pointsin the profile is (weakly) nonlinearly unstable. Thisis an instability in the full nonlinear Euler equationsthat is not restricted to infinitesimal disturbances.This observation is consistent with experimentalevidence that shear flow in a channel with a linearprofile (so-called plane Couette flow) is in fact un-stable at high Reynolds numbers, despite mathe-matical results indicating stability. This is the sameparadox that we discussed at the beginning of thisarticle in connection with Reynolds’s experiment.Again, it is likely that turbulence appears as a con-sequence of instability to small but finite distur-bances that generate vorticity.

Instability of More General Euler FlowsWe now turn to the problem of examining Spec Lin the case of more general fluid flows U0(x). Theconstruction of explicit eigenvalues is then a daunt-ing task on which little progress has been made.However, it has proved possible in many cases toobtain certain information about the continuousspectrum. In a series of papers (for example, [9])

Friedlander and Vishik used high-frequency as-ymptotics and techniques of geometric optics toobtain a sufficient condition for instability with re-spect to growth in the energy norm. They show thata quantity Λ that can be viewed as a “fluid Lya-punov exponent” gives a lower bound on thegrowth rate of solutions to the linearized Eulerequations. Classically a Lyapunov exponent asso-ciated with a dynamical system is the exponentialgrowth rate of a tangent vector that is moved andstretched/contracted by the flow field of the sys-tem. The fluid Lyapunov exponent is also deter-mined by a system of ODEs over the trajectoriesof the fluid velocity U0(x).

Conceptually we envision the following situa-tion. Take a point x0 in the fluid. At this point wemake an initial disturbance in the following form:

(13) v∣∣∣t=0

= b0eiξξξ0/ε,

where the amplitude b0 is a positive function ofsmall support sharply peaked at x0. The parame-ter ε is very small, so the disturbance oscillatesrapidly in space. This “wavelet” is moved by theflow to a point x(t) so that at a time t later the dom-inant wave vector is ξξξ(t)

/ε and the amplitude has

evolved to a vector b(t). This is illustrated in Fig-ure 6. The evolution with time of b(t) and ξξξ(t) alongthe trajectories of U0(x) is governed by a specificcoupled system of ODEs. The specific equations canbe found in [9]. The fluid Lyapunov exponent Λ isdefined to be

(14) Λ = limt→∞

1t

log supx0,ξ0,b0

(ξ0·b0)=0|ξξξ0|=1,|b0|=1

∣∣∣b(x0,ξξξ0, t)∣∣∣.

Figure 6. Evolution of high-frequency wavelet disturbance.

Sou

rce:

Su

san

Fri

edla

nd

er, P

ark C

ity/

AM

SSe

ries

, vol.

5, 1

999.



The existence of this strict limit is justified by Os-eledets’s Ergodic Theorem. It is proved by Vishikand Friedlander (1993) that the positivity of Λ forany given steady flow U0(x) implies the existenceof a nonempty unstable continuous spectrum forthe operator L associated with U0. This result isproved for the spectrum in the space L2 for flowsin arbitrary physical space dimension and withperodic or free space boundary conditions. In manyparticular examples of U0 it is possible to use thecoupled system of ODE for ξξξ(t) and b(t) to calcu-late a lower bound on Λ and hence prove that theflow U0 is unstable. We note that it is possiblethat Λ is zero and yet the flow is still unstable be-cause of the existence of discrete unstable eigen-values. This is in fact the situation for plane par-allel shear flow discussed earlier in this article.

Recently Vishik (1996) sharpened the above re-sult to prove, in the case of spatially periodicboundary conditions, that Λ gives exactly the max-imal growth rate in the continuous spectrum of L,i.e., Λ defines the essential spectral radius of theevolution operator exp L in the space L2.

To detect instabilities using Λ , it is sufficientto make a particular “intelligent” choice for the ini-tial position and direction of the wavelet that islikely to give an exponentially growing amplitudeb(t). For example, consider the situation wherethere exists a point xh at which U0(x) has a hy-perbolic stagnation (fixed) point; i.e., U0(xh) = 0and the flow lines in the neighborhood of xh arehyperbolic in shape. In such a situation we choose

x0 = xh. It is then easy to solve the system of ODEs,which become constant coefficient equations overthe trajectories through the fixed point of U0, andhence demonstrate the existence of an exponen-tially growing amplitude vector b. This is a specialcase of a more general result proved in Friedlan-der and Vishik [9] that any fluid flow with expo-nential stretching (i.e., positive classical Lyapunovexponent) has a positive fluid exponent Λ andhence is fluid-dynamically unstable. This confirmsan observation originally made by Arnold [5] thatexponential stretching is related to fluid instabil-ities.

One interesting specific example of a fluid ve-locity U0(x) satisfying equation (6) that contains ex-ponential stretching and hence instability is a so-called “ABC” flow, where the three components ofU0 in Cartesian coordinates are

U01 ≡dx1

dt= A sinx3 + C cosx2,

U02 ≡dx2

dt= B sinx1 +A cosx3,(15)

U03 ≡dx3

dt= C sinx2 + B cosx1.

This flow is named after the three mathematiciansArnold, Beltrami, and Childress, who have con-tributed much to our understanding and appreci-ation of classes of “chaotic” flows of which (15) isan example. For nonzero values of the constantsA , B, and C the system (15) is not globally inte-grable (Arnold [1965]), Henon [1966]). The topol-ogy of the flow lines is very complicated and canonly be investigated numerically to reveal regionsof chaotic behavior as illustrated, for example, bythe results of Dombre et al. (1986).

Although exponential stretching is a sufficientcondition for Λ to be positive, it is not necessary.There are examples of fluid flows U0(x) that haveintegrals of the motion and where all the classicalLyapunov exponents are zero and yet Λ is positive[9]. Such an example is a model for a vortex ringwhere the flow lines wrap ergodically around thesurfaces of nested tori. Under certain curvatureconditions on the geometry of these flow lines, itcan be shown that Λ is small but positive andhence the vortex ring is an unstable fluid flow. Ex-periments show instability of a vortex ring mani-fested by wavy perturbations on the outer shell ofthe ring.

Our discussion of the instability of general flowshas concerned only the linearized equations (7)–(8).However, we stressed earlier in the article the im-portance of the nonlinearity of the Euler equa-tions. Recently Friedlander, Strauss, and Vishik [7]proved that for large classes of evolution PDEslinear instability implies nonlinear instability undercertain rather general conditions. This abstracttheorem can be applied to the Euler equations toprove nonlinear instability in the Sobolev space Hs,

Figure 7. Detail of a Poincaré section of an ABC flow showing ahyperbolic point xh.



s > n2 + 1, of flows U0(x) where it is known that

SpecL contains a “gap” that allows separation ofa growing solution to (7)–(8). To date we do notknow enough about the detailed structure of SpecLto be sure that the gap condition always holds forany U0(x). However, in some specific exampleswhere the physical space dimension n is 2, it is pos-sible to verify the gap condition. This includesplane shear flows with oscillating profiles, whichare therefore nonlinearly unstable in Hs, s > 2.

We briefly mention that there is a considerablenumber of mathematicians who have studied as-pects of the continuous unstable spectrum arisingin the equations of fluid dynamics. Constraints ofspace prevent our mentioning many beautiful re-sults, so we just give a few examples that an in-terested reader could pursue: the high frequencyasymptotics applied to sound waves by F. G. Fried-lander (1958), the pioneering work of Eckhoff(1981) on local instabilities in hyperbolic systems,the study of the continuous spectrum in idealmagnetohydrodynamics of Hameiri (1985) or Lif-schitz (1989), the treatment of hydrodynamic sta-bility “without eigenvalues” given by Trefethen etal. (1993).

Concluding RemarksThe thrust of our discussion indicates that in theabsence of stabilizing forces, there is an underly-ing tendency to instability for almost all inviscidfluid flows. The nature of the instability will varywith the structure of the flow and the quantity thatmeasures the growth of the disturbance. Thegrowth rate may be fast or slow. The visible effectsof the instability on the “pattern” of the fluid canbe very different. The observed pattern of an un-stable flow may evolve gradually, as with a sinewave that rolls up into vortices, or it may changeviolently, as when a laminar flow becomes abruptlyturbulent. This suggests the need for a definitionof a “scale” of instabilities. At present theoreticalresults give us just hints as to what is happeningin experiments and in nature.

AcknowledgmentsWe thank the editors of the Notices for their care-ful reading of this article and their beneficial sug-gestions. We thank Loretta Allen for her excellent“emergency” typing, which enabled us to meet ourdeadline. We thank the Princeton mathematics de-partment for their hospitality during their pro-gram on the Euler equations, which allowed the au-thors to discuss this article together in person.

Skeletal Bibliography[1] O. REYNOLDS, An experimental investigation of the cir-

cumstances which determine whether the motion ofwater shall be direct or sinuous, and the law of re-sistance in parallel channels, Philos. Trans. Roy. Soc.London 174 (1883), 935–982.

[2] C. GODRECHE and P. MANNEVILLE (eds.), Hydrodynam-ics and Nonlinear Instability, Cambridge UniversityPress, London and New York, 1998.

[3] P. CHOSSAT and G. IOOSS, The Couette-Taylor Problem,Springer-Verlag, Berlin, 1994.

[4] P. CONSTANTIN, A few results and open problems re-garding incompressible fluids, Notices Amer. Math.Soc. 42 (1995), 658–663.

[5] V. I. ARNOLD, Notes on the three-dimensional flow pat-tern of a perfect fluid in the presence of a small per-turbation of the initial velocity field, Appl. Math.Mech. 36 (1972), 236–242.

[6] V. I. YUDOVICH, The Linearization Method in Hydro-dynamical Stability Theory, Transl. Math. Mono-graphs, vol. 74, Amer. Math. Soc., Providence, RI,1984.

[7] S. FRIEDLANDER, W. STRAUSS, and M. VISHIK, Nonlinearinstability in an ideal fluid, Ann. Inst. H. PoincaréAnal. Non Linéaire 14 (1997), 187–209.

[8] V. I. YUDOVICH, Loss of smoothness and intrinsic in-stability of an ideal fluid, preprint VINITI no. 2331-B98, Chaos, (to appear).

[9] S. FRIEDLANDER and M. VISHIK, Instability criteria forsteady flows of a perfect fluid, Chaos 2 (1992),455–460.

About the CoverOn the cover is a photograph obtained

through direct numerical simulation of “tur-bulence” arising in the two-dimensional Navier-Stokes equations with periodic boundary con-ditions and initial random distribution. Thepicture is a visualization of the magnitude ofthe vorticity field at an instant in time. Vor-tices correspond to elliptical regions of theflow where rotation dominates strain. Thewell-mixed background flow corresponds tohyperbolic regions where strain dominates ro-tation.

The computations were performed by MarieFarge of the Laboratoire de Météorologie Dy-namique du CNRS at the École NormaleSupérieure de Paris. The visualization wasdone in collaboration with Jean FrancoisColonna of LACTAMME at the École Polytech-nique. Thanks go to them for the use of theirphotograph.

—S. F.


susan friedlander and victor yudovich- instabilities in fluid motion

Documents